Question

If a and ß are the zeroes of the polynomial x -5x +k such that alfa - Beta=1
Find the value of k.​

Answer 1

P(x) = x²-5x+k

Alpha and Beta are zeroes of the polynomial.

We know from the relations of the coefficients and Zeroes of the polynomial that :-

Sum of the zeroes = - b/a

Hence

In this case :-

$\alpha + \beta = 5$

Also, It is give that :-

$\alpha - \beta = 1 \\ \\ \\ \alpha = 1 + \beta$

Putting down this in first equation

$1 + \beta + \beta = 5 \\ \\2 \beta = 4 \\ \\ \\ \beta = 2 \\ \\ \\ \alpha = 1 + \beta = 3$

Hence,

Product of zeroes will be :-

$\alpha \beta = \frac{c}{a} \\ \\ \\k = 2(3) \\ \\k = 6$

Hence, Value of k is 6

Answer 2

GIVEN :

α and β are zeroes of the polynomial x² - 5x + k

α - β = 1 ----(1)

From the relation between zeroes and coefficients,

Sum of zeroes (α + β) = -b/a = -(-5)/1 = 5

(α + β) = 5 ------(2)

Solve both eq - (1) and (2)

α - β = 1

α + β = 5

--------------

2α = 6

--------------

α = 6/3

α = 2

Substitute α in eq - (2)

α + β = 5

2 + β = 5

β = 5 - 2

β = 3

Thus, α and β are 2 and 3

Then,

Product of zeroes = αβ = k = 2 × 3 = 6

Therefore, the value of k is 6.